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The Donnelly-Fefferman theorem on \(q\)-pseudoconvex domains. (English) Zbl 1214.32015
The authors extend the notion of a \(q\)-subharmonic function introduced in the \(C^{2}\) class by L.-H. Ho [Math. Ann. 290, No. 1, 3–18 (1991; Zbl 0714.32006)] to the class of upper semicontinuous functions, and use these functions to obtain \(L^{2}\)-estimations of Donnelly-Fefferman type for the \(\overline{\partial}\)-equations on certain domains with non-smooth boundaries. More precisely, let \(\varphi\) be an upper semicontinuous function on an open set in \(\mathbb{C}^{n}\); it is called \(q\)-subharmonic in \(U\) if, for every linear complex space \(H\) of dimension \(q\), and for every compact set \(K\subset U\cap H\), it has the following property: if \(h\) is a continuous harmonic function on \(K\) and \(h\leq\varphi\) on \(\partial K\), then \(h\leq\varphi\) on \(K\). Plurisubharmonicity is equivalent to 1-subharmonicity; an open set \(D\) on \(\mathbb{C}^{n}\) is called \(q\)-pseudoconvex if there is a \(q\)-subharmonic exhaustion function for \(D\).
The main result is the following: Let \(D\) be a \(q\)-pseudoconvex domain in \(\mathbb{C}^{n}\), let \(\varphi\) be a given \(q\)-subharmonic function in \(D\), and let \(\psi\in C^{2}(D)\) be strictly plurisubharmonic such that \(-e^{-\psi}\) is \(q\)-subharmonic. Further, let \(\varepsilon\in(0,1)\). Then for every \(\overline{\partial}\)-closed \((0,r)\)-form \(g\), with \(q\leq r\leq n\), there is a solution of \(\overline{\partial}u=g\) such that
\[ \int_{D}|u|^{2}e^{-\varphi+\varepsilon\psi}dV\leq\frac{4}{\varepsilon(1-\varepsilon)^{2}}\frac{1}{r}\mathop{\sum{'}}\limits_{| K|=r-1}\mathop{\sum}\limits_{j,k}\int_{D}\psi^{j\overline{k}}g_{jK}\overline{g}_{kK}e^{-\varphi+\varepsilon\psi}dV\tag{*} \]
whenever the right hand side of \((*)\) is bounded. Here, \((\psi^{j\overline{k}})=(\psi_{\nu\overline{\mu}})^{-1}\).
As a corollary, the authors prove the following approximation result: Let \(D\) be as above, \(h\) a continuous \(q\)-subharmonic function in \(D\), and \(K=\left\{ z\in D: h(z)\leq0\right\}\subset\subset D\). If an \(\overline{\partial}\)-closed \((0,r)\)-form \(f\), \(r\geq q-1\), is smooth in a neighborhood of \(K\), then, for each \(\delta>0\), there is a \(\overline{\partial}\)-closed \((0,r)\)-form \(g_{\delta}\) with coefficients in \(L^{2}(D)\) such that \(\| f-g_{\delta}\|_{L^{2}(K)}<\delta\).
The proof of the main result is done in two steps: first, the authors treat the case of a smoothly bounded \(q\)-pseudoconvex domain \(\Omega\) and \(\varphi\), \(\psi\) smooth up to \(\overline{\Omega}\), \(\psi\) positive definite and \(-e^{-\psi}\) \(q\)-subharmonic using the Bochner-Kodaira identity. The general case is approached by approximating \(D\) by an increasing sequence \(\left\{ D_{\nu}\right\}\), \(D_{\nu}\subset\subset D\), and choosing a decreasing sequence \(\left\{ \varphi_{\mu}\right\}\) of smooth \(q\)-subharmonic functions which converges pointwise to \(\varphi\), and applying the estimate obtained before, with \(\varphi_{\nu}\), \(w=e^{-\varepsilon\psi}\) and \(\Omega=D_{\nu}\).

32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
32F10 \(q\)-convexity, \(q\)-concavity
Full Text: Euclid
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